DC Branches
DC Branches represent overhead lines and cables in DC connections and DC grids.
Parameters
Set of parameters used to model DC branches as defined in the input data
| name | symb. | unit | type | default | definition |
|---|---|---|---|---|---|
| index | $d$ | - | Int | - | unique index of the DC branch |
| f_bus | $i$ | - | Int | - | unique index of the bus to which the DC branch is originating from |
| t_bus | $j$ | - | Int | - | unique index of the bus to which the DC branch is terminating at |
| r | $r_{dc}$ | p.u. | Real | - | resistance of the DC branch |
| l | $l_{dc}$ | p.u. | Real | - | inductance of the DC branch (not used) |
| c | $c_{dc}$ | p.u. | Real | - | capacitance of the DC branch (not used) |
| rateA | $\overline{P_{dc}}$ | MW | Real | - | long term rating of the DC branch |
| rateB | $\overline{P^{st}_{dc}}$ | MW | Real | - | short term term rating of the DC branch |
| rateC | $\overline{P^{em}_{dc}}$ | MW | Real | - | emergency rating of the DC branch |
| status | $\delta_{dc}$ | - | Int | - | status indicator of the DC branch |
| p | $n_{p}$ | - | Int | - | number of DC poles, specified at test case level |
| cost | $C_{dc}$ | - | Real | - | investment cost for DC branch used in TNEP problems |
Variables
Optimisation variables representing DC branch behaviour
| name | symb. | unit | formulation | definition |
|---|---|---|---|---|
| p_dcgrid | $P_{d,i,j}$ | p.u. | ACP, ACR, LPAC, IVR, SOC, DCP, NF | Active power flow through DC branch d, connecting DC nodes i and j |
| igrid_dc | $I_{d,i,j}$ | p.u. | IVR | Current flow through DC branch d, connecting DC nodes i and j |
| ccm_dcgrid | $J_{d,i,j}$ | p.u. | SOC, QC | Square of current flow through DC branch d, connecting DC nodes i and j |
| wdcr | $W_{d,i,j}$ | p.u. | SOC, QC | Lifted variable representing bilinear voltage product ($V_{i} \cdot V_{j}$) of branch d, connecting DC nodes i and j |
Constraints
Flow, current and voltage product limits
\[\begin{align} - \overline{P_{dc}} &\leq P_{d,i,j} \leq \overline{P_{dc}} \\ - \overline{I_{dc}} &\leq I_{d,i,j} \leq \overline{I_{dc}} \\ 0 &\leq J_{d,i,j} \leq \overline{I_{dc}^{2}} \\ 0 &\leq W_{d,i,j} \leq max(V_{i}^{2}, V_{j}^{2}) \end{align}\]
DC branch admittance
\[g_{dc} = \frac{1}{r_{dc}}\]
Ohm's law
ACP, ACR model:
\[\begin{align} P_{d,i,j} &= n_{p} \cdot g_{dc} \cdot V_{i} \cdot (V_{i} - V_{j}) \\ P_{d,j,i} &= n_{p} \cdot g_{dc} \cdot V_{j} \cdot (V_{j} - V_{i}) \end{align}\]
SOC, QC bus injection model (BIM):
\[\begin{align} P_{d,i,j} &= n_{p} \cdot g_{dc} \cdot (W_{i} - W_{d,i,j}) \\ P_{d,j,i} &= n_{p} \cdot g_{dc} \cdot (W_{j} - W_{d,i,j}) \end{align}\]
SOC, QC branch flow model (BFM):
\[\begin{align} W_{j} &= W_{i} - \frac{2 \cdot r_{dc} \cdot P_{d,i,j}} {n_{p}} + r_{dc}^{2} \cdot J_{d,i,j} \\ P_{d,i,j} + P_{d,j,i} &= r_{dc} \cdot n_{p} \cdot J_{d,i,j} \\ P_{d,i,j}^{2} &\leq n_{p}^{2} \cdot W_{i} \cdot J_{d,i,j} \end{align}\]
IVR model:
\[\begin{align} V_{j} = V_{i} - \frac{r_{dc} \cdot I_{d,i,j}}{n_{p}} \\ V_{i} = V_{j} - \frac{r_{dc} \cdot I_{d,j,i}}{n_{p}} \\ P_{d,i,j} = I_{d,i,j} \cdot V_{i} \\ P_{d,j,i} = I_{d,j,i} \cdot V_{j} \end{align}\]
DCP and NF model:
In this model there are no losses, as such only the active power limits are binding.